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Properties

Label 588.2.f.a.293.1
Level $588$
Weight $2$
Character 588.293
Analytic conductor $4.695$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [588,2,Mod(293,588)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("588.293"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(588, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 1])) N = Newforms(chi, 2, names="a")
 
Level: \( N \) \(=\) \( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 588.f (of order \(2\), degree \(1\), minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [2,0,-3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(4.69520363885\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 84)
Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label 293.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 588.293
Dual form 588.2.f.a.293.2

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(-1.50000 - 0.866025i) q^{3} +3.00000 q^{5} +(1.50000 + 2.59808i) q^{9} +5.19615i q^{11} +(-4.50000 - 2.59808i) q^{15} +3.00000 q^{17} -1.73205i q^{19} +5.19615i q^{23} +4.00000 q^{25} -5.19615i q^{27} -1.73205i q^{31} +(4.50000 - 7.79423i) q^{33} +7.00000 q^{37} +6.00000 q^{41} +4.00000 q^{43} +(4.50000 + 7.79423i) q^{45} +3.00000 q^{47} +(-4.50000 - 2.59808i) q^{51} -5.19615i q^{53} +15.5885i q^{55} +(-1.50000 + 2.59808i) q^{57} -3.00000 q^{59} +12.1244i q^{61} +5.00000 q^{67} +(4.50000 - 7.79423i) q^{69} -10.3923i q^{71} -12.1244i q^{73} +(-6.00000 - 3.46410i) q^{75} -1.00000 q^{79} +(-4.50000 + 7.79423i) q^{81} -12.0000 q^{83} +9.00000 q^{85} -9.00000 q^{89} +(-1.50000 + 2.59808i) q^{93} -5.19615i q^{95} -6.92820i q^{97} +(-13.5000 + 7.79423i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 3 q^{3} + 6 q^{5} + 3 q^{9} - 9 q^{15} + 6 q^{17} + 8 q^{25} + 9 q^{33} + 14 q^{37} + 12 q^{41} + 8 q^{43} + 9 q^{45} + 6 q^{47} - 9 q^{51} - 3 q^{57} - 6 q^{59} + 10 q^{67} + 9 q^{69} - 12 q^{75}+ \cdots - 27 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/588\mathbb{Z}\right)^\times\).

\(n\) \(197\) \(295\) \(493\)
\(\chi(n)\) \(-1\) \(1\) \(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) −1.50000 − 0.866025i −0.866025 − 0.500000i
\(4\) 0 0
\(5\) 3.00000 1.34164 0.670820 − 0.741620i \(-0.265942\pi\)
0.670820 + 0.741620i \(0.265942\pi\)
\(6\) 0 0
\(7\) 0 0
\(8\) 0 0
\(9\) 1.50000 + 2.59808i 0.500000 + 0.866025i
\(10\) 0 0
\(11\) 5.19615i 1.56670i 0.621582 + 0.783349i \(0.286490\pi\)
−0.621582 + 0.783349i \(0.713510\pi\)
\(12\) 0 0
\(13\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(14\) 0 0
\(15\) −4.50000 − 2.59808i −1.16190 − 0.670820i
\(16\) 0 0
\(17\) 3.00000 0.727607 0.363803 − 0.931476i \(-0.381478\pi\)
0.363803 + 0.931476i \(0.381478\pi\)
\(18\) 0 0
\(19\) − 1.73205i − 0.397360i −0.980064 − 0.198680i \(-0.936335\pi\)
0.980064 − 0.198680i \(-0.0636654\pi\)
\(20\) 0 0
\(21\) 0 0
\(22\) 0 0
\(23\) 5.19615i 1.08347i 0.840548 + 0.541736i \(0.182233\pi\)
−0.840548 + 0.541736i \(0.817767\pi\)
\(24\) 0 0
\(25\) 4.00000 0.800000
\(26\) 0 0
\(27\) − 5.19615i − 1.00000i
\(28\) 0 0
\(29\) 0 0 1.00000 \(0\)
−1.00000 \(\pi\)
\(30\) 0 0
\(31\) − 1.73205i − 0.311086i −0.987829 − 0.155543i \(-0.950287\pi\)
0.987829 − 0.155543i \(-0.0497126\pi\)
\(32\) 0 0
\(33\) 4.50000 − 7.79423i 0.783349 − 1.35680i
\(34\) 0 0
\(35\) 0 0
\(36\) 0 0
\(37\) 7.00000 1.15079 0.575396 − 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(38\) 0 0
\(39\) 0 0
\(40\) 0 0
\(41\) 6.00000 0.937043 0.468521 − 0.883452i \(-0.344787\pi\)
0.468521 + 0.883452i \(0.344787\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 − 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 0 0
\(45\) 4.50000 + 7.79423i 0.670820 + 1.16190i
\(46\) 0 0
\(47\) 3.00000 0.437595 0.218797 − 0.975770i \(-0.429787\pi\)
0.218797 + 0.975770i \(0.429787\pi\)
\(48\) 0 0
\(49\) 0 0
\(50\) 0 0
\(51\) −4.50000 − 2.59808i −0.630126 − 0.363803i
\(52\) 0 0
\(53\) − 5.19615i − 0.713746i −0.934153 − 0.356873i \(-0.883843\pi\)
0.934153 − 0.356873i \(-0.116157\pi\)
\(54\) 0 0
\(55\) 15.5885i 2.10195i
\(56\) 0 0
\(57\) −1.50000 + 2.59808i −0.198680 + 0.344124i
\(58\) 0 0
\(59\) −3.00000 −0.390567 −0.195283 − 0.980747i \(-0.562563\pi\)
−0.195283 + 0.980747i \(0.562563\pi\)
\(60\) 0 0
\(61\) 12.1244i 1.55236i 0.630509 + 0.776182i \(0.282846\pi\)
−0.630509 + 0.776182i \(0.717154\pi\)
\(62\) 0 0
\(63\) 0 0
\(64\) 0 0
\(65\) 0 0
\(66\) 0 0
\(67\) 5.00000 0.610847 0.305424 − 0.952217i \(-0.401202\pi\)
0.305424 + 0.952217i \(0.401202\pi\)
\(68\) 0 0
\(69\) 4.50000 − 7.79423i 0.541736 − 0.938315i
\(70\) 0 0
\(71\) − 10.3923i − 1.23334i −0.787222 − 0.616670i \(-0.788481\pi\)
0.787222 − 0.616670i \(-0.211519\pi\)
\(72\) 0 0
\(73\) − 12.1244i − 1.41905i −0.704681 − 0.709524i \(-0.748910\pi\)
0.704681 − 0.709524i \(-0.251090\pi\)
\(74\) 0 0
\(75\) −6.00000 − 3.46410i −0.692820 − 0.400000i
\(76\) 0 0
\(77\) 0 0
\(78\) 0 0
\(79\) −1.00000 −0.112509 −0.0562544 − 0.998416i \(-0.517916\pi\)
−0.0562544 + 0.998416i \(0.517916\pi\)
\(80\) 0 0
\(81\) −4.50000 + 7.79423i −0.500000 + 0.866025i
\(82\) 0 0
\(83\) −12.0000 −1.31717 −0.658586 − 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) 0 0
\(85\) 9.00000 0.976187
\(86\) 0 0
\(87\) 0 0
\(88\) 0 0
\(89\) −9.00000 −0.953998 −0.476999 − 0.878904i \(-0.658275\pi\)
−0.476999 + 0.878904i \(0.658275\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −1.50000 + 2.59808i −0.155543 + 0.269408i
\(94\) 0 0
\(95\) − 5.19615i − 0.533114i
\(96\) 0 0
\(97\) − 6.92820i − 0.703452i −0.936103 − 0.351726i \(-0.885595\pi\)
0.936103 − 0.351726i \(-0.114405\pi\)
\(98\) 0 0
\(99\) −13.5000 + 7.79423i −1.35680 + 0.783349i
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 588.2.f.a.293.1 2
3.2 odd 2 588.2.f.c.293.1 2
4.3 odd 2 2352.2.k.d.881.2 2
7.2 even 3 588.2.k.d.521.1 2
7.3 odd 6 588.2.k.c.509.1 2
7.4 even 3 84.2.k.b.5.1 yes 2
7.5 odd 6 84.2.k.a.17.1 yes 2
7.6 odd 2 588.2.f.c.293.2 2
12.11 even 2 2352.2.k.a.881.2 2
21.2 odd 6 588.2.k.c.521.1 2
21.5 even 6 84.2.k.b.17.1 yes 2
21.11 odd 6 84.2.k.a.5.1 ✓ 2
21.17 even 6 588.2.k.d.509.1 2
21.20 even 2 inner 588.2.f.a.293.2 2
28.11 odd 6 336.2.bc.b.257.1 2
28.19 even 6 336.2.bc.d.17.1 2
28.27 even 2 2352.2.k.a.881.1 2
35.4 even 6 2100.2.bi.e.1601.1 2
35.12 even 12 2100.2.bo.a.1949.2 4
35.18 odd 12 2100.2.bo.f.1349.1 4
35.19 odd 6 2100.2.bi.f.101.1 2
35.32 odd 12 2100.2.bo.f.1349.2 4
35.33 even 12 2100.2.bo.a.1949.1 4
63.4 even 3 2268.2.bm.f.593.1 2
63.5 even 6 2268.2.w.a.269.1 2
63.11 odd 6 2268.2.w.f.1349.1 2
63.25 even 3 2268.2.w.a.1349.1 2
63.32 odd 6 2268.2.bm.a.593.1 2
63.40 odd 6 2268.2.w.f.269.1 2
63.47 even 6 2268.2.bm.f.1025.1 2
63.61 odd 6 2268.2.bm.a.1025.1 2
84.11 even 6 336.2.bc.d.257.1 2
84.47 odd 6 336.2.bc.b.17.1 2
84.83 odd 2 2352.2.k.d.881.1 2
105.32 even 12 2100.2.bo.a.1349.1 4
105.47 odd 12 2100.2.bo.f.1949.1 4
105.53 even 12 2100.2.bo.a.1349.2 4
105.68 odd 12 2100.2.bo.f.1949.2 4
105.74 odd 6 2100.2.bi.f.1601.1 2
105.89 even 6 2100.2.bi.e.101.1 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
84.2.k.a.5.1 ✓ 2 21.11 odd 6
84.2.k.a.17.1 yes 2 7.5 odd 6
84.2.k.b.5.1 yes 2 7.4 even 3
84.2.k.b.17.1 yes 2 21.5 even 6
336.2.bc.b.17.1 2 84.47 odd 6
336.2.bc.b.257.1 2 28.11 odd 6
336.2.bc.d.17.1 2 28.19 even 6
336.2.bc.d.257.1 2 84.11 even 6
588.2.f.a.293.1 2 1.1 even 1 trivial
588.2.f.a.293.2 2 21.20 even 2 inner
588.2.f.c.293.1 2 3.2 odd 2
588.2.f.c.293.2 2 7.6 odd 2
588.2.k.c.509.1 2 7.3 odd 6
588.2.k.c.521.1 2 21.2 odd 6
588.2.k.d.509.1 2 21.17 even 6
588.2.k.d.521.1 2 7.2 even 3
2100.2.bi.e.101.1 2 105.89 even 6
2100.2.bi.e.1601.1 2 35.4 even 6
2100.2.bi.f.101.1 2 35.19 odd 6
2100.2.bi.f.1601.1 2 105.74 odd 6
2100.2.bo.a.1349.1 4 105.32 even 12
2100.2.bo.a.1349.2 4 105.53 even 12
2100.2.bo.a.1949.1 4 35.33 even 12
2100.2.bo.a.1949.2 4 35.12 even 12
2100.2.bo.f.1349.1 4 35.18 odd 12
2100.2.bo.f.1349.2 4 35.32 odd 12
2100.2.bo.f.1949.1 4 105.47 odd 12
2100.2.bo.f.1949.2 4 105.68 odd 12
2268.2.w.a.269.1 2 63.5 even 6
2268.2.w.a.1349.1 2 63.25 even 3
2268.2.w.f.269.1 2 63.40 odd 6
2268.2.w.f.1349.1 2 63.11 odd 6
2268.2.bm.a.593.1 2 63.32 odd 6
2268.2.bm.a.1025.1 2 63.61 odd 6
2268.2.bm.f.593.1 2 63.4 even 3
2268.2.bm.f.1025.1 2 63.47 even 6
2352.2.k.a.881.1 2 28.27 even 2
2352.2.k.a.881.2 2 12.11 even 2
2352.2.k.d.881.1 2 84.83 odd 2
2352.2.k.d.881.2 2 4.3 odd 2